The heart of the Back Bay (Newbury/boylston st.) tends to be a very fancy, very "haughty" neighborhood, which tends towards fancy, overpriced places (with the exception of Cafe Jaffa, below). However, here are some of my favorites, within walking distance of Copley Square (as a landmark).
There's a wonderful Ethiopian restaurant in the South End, which is about a 10 minute walk from Back Bay. It's called Addis Red Sea restaurant (http://www.addisredsea.com). We eat there all the time, as we used to live around the corner, and it's across the street from a theater we attend often. Very nice atmosphere, and the food is great. Some of it is spicy, but they can steer you away from the really hot stuff, if you don't like that. If you go, make sure to try the Ethiopian honey wine, and/or the dark beer.
For Israeli/Middle Eastern food, I like Caffe Jaffa (Cafe Jaffa
48 Gloucester Street
Boston --Gloucester runs between Newbury and Boylston st, in the Back Bay.
The shwarma and the cold appetizers (humus/tabolueh/grape leaves) are really good, and the portions are ample, and it's cheap :)
In the Fenway neighborhood, which is next to Back Bay (near Fenway Park, about a 15 minute walk) are two great places right across the street from each other on Jersey St: Brown Sugar Cafe, which is Thai food, and Buteco, which is Brazilian food. Both have fantastic food, and you'd waddle out of either place very happy.
130 Jersey Street
Boston, Massachusetts 02215-5117 ph: 617.247.9508
For Indian food, you should try the India Quality Restaurant, in Kenmore Sq., which is just at the edge of the Back Bay, an easy walk. This is my favorite Indian restaurant in Boston proper. http://indiaqualityrest.com/
As you can probably tell, I like to eat, so if you want more suggestions, let me know. :)